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Averages of shifted convolution sums for arithmetic functions

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Abstract

Let f be a full-level cusp form for GLm(ℤ) with Fourier coefficients Af(cm-2,..., c1, n): Let λ(n) be either the von Mangoldt function Λ(n) or the k-th divisor function τk(n): We consider averages of shifted convolution sums of the type Σ|h|⩽H | ΣX<n⩽2XAf (1,..., 1, n+h)λ(n)|2. We succeed in obtaining a saving of an arbitrary power of the logarithm, provided that \(X{\frac{8}{33}+\epsilon}\) ⩽ H ⩽ X1-ɛ.

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Acknowledgements

The author was very grateful to the referees for careful reading of the paper and useful suggestions. This work was supported in part by the National Natural Science Foundation of China (Grant Nos. 11771252, 11531008).

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  1. School of Mathematics, Shandong University, Jinan, 250100, China

    Miao Lou

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  1. Miao Lou
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Correspondence to Miao Lou.

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Lou, M. Averages of shifted convolution sums for arithmetic functions. Front. Math. China 14, 123–134 (2019). https://doi.org/10.1007/s11464-019-0749-9

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  • DOI: https://doi.org/10.1007/s11464-019-0749-9

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